Magnetic Vector Potential

The divergence of a magnetic field is zero, and the divergence of the curl of any vector is always zero. So, a magnetic vector potential can be defined such that its curl equals the magnetic field.

The differential form of Ampere's Law states that the curl of the magnetic field equals the vacuum permeability times the current density.

Replacing the magnetic field with the magnetic vector potential, the equation gets modified.

Now, according to the vector product rule, the curl of the curl of a vector equals the gradient of its divergence minus its vector Laplacian.

Since the vector potential can be chosen to be non divergent, the Laplacian of the magnetic vector potential equals the permeability times the current density.

This differential equation bears an analogy to Poisson's equation in electrostatics.

Considering that the current density goes to zero at infinity, the solution of the Laplacian equation gives the vector potential.

The magnetic field due to any current source can be evaluated using the corresponding vector potential.